On invariants of null curves in the pseudo-Euclidean geometry

Let M(n, p) be the group of all transformations of an n-dimensional pseudo-Euclidean space E-p(n) of index p generated by all pseudo-orthogonal transformations and parallel translations of E-p(n). Definitions of a pseudo-Euclidean type of a null curve, an invariant parametrization of a null curve and an M(n, p)-equivalence of curves are introduced. All possible invariant parametrizations of a null curve with a fixed pseudo-Euclidean type are described. The problem of the M(n, p)-equivalence of null curves is reduced to that of null paths. Global conditions of the M(n, p)-equivalence of null curves are given in terms of the pseudo-Euclidean type of a null curve and the system of polynomial differential M(n, p)-invariants of a null curve x(s). (C) 2011 Elsevier B.V. All rights reserved.